Question

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Intersecting the y-axis at a distance of 2 units above the origin and making an
angle of 30° with positive direction of the x-axis.​

Answer 1

Step-by-step explanation:

if two lines intersect at point ( a,b) it means point ( a,b) satisfy equation of both straight lines. So you can put one by one in both equation to get value of c ( intercept)

line intersect at ( 0, 2) i.e point ( 0, 2 ) satisfy the equation y=mx+c

2=m*0 + c

c=2

slope(m) = tan 30° = 1/√3

putting value of m and c in equation of straight line

y=1x/√3 + 2

√3y=x +2√3

x - √3y +2√3=0

Answer 2

Given Question :-

• Find the equation of line which intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30° with positive direction of the x-axis.

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Understanding the concept used :-

In this question, we use the slope intercept form to find the equation of line as we have provided the intercept on y axis as well slope ( can be evaluated using tanθ).

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Formula Used

☆ Slope-Intercept Form of the Equation of a Line :-

☆ The linear equation written in the form :- y = m x + c. where:

• m is the slope
• c is the y-intercept.

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Solution :-

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$\begin{gathered}\begin{gathered}\bf Given - \begin{cases} &\sf{intercept \: on \: y \: axis \: = 2 \: units} \\ &\sf{angle \:which \: line \: makes \: with \: + ve \: x \: axis \: = 30°} \end{cases}\end{gathered}\end{gathered}$

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$\tt \:  ⟼Since \: line \: makes \: an \: angle \: of \: 30° \: with \: \\ \tt \:  positive \: direction \: of \: x \: - \: axis.$

$\tt\implies \: \: Slope \: of \: line, \:m \: = tan30° \: = \dfrac{1}{ \sqrt{3} }$

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$\tt \:  ⟼Also, \: line \: intersects \: the \: y - axis \: at a \: \\ \tt \:  distance \: of \: 2 \: units \: above \: the \: origin.$

$\tt\implies \:intercept \: on \: y \: - \: axis, \: c \: = 2$

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$\tt \:  ⟼ \blue{So, \: equation \: of \: line \: is \: given \: by}$

$\tt \:  ⟼ \red{y \: = mx \: + \: c}$

☆ On substituting the values of m and c, we get

$\tt \:  ⟼ \: y \: = \dfrac{1}{ \sqrt{3} } x + 2$

$\tt \:  ⟼ \sqrt{3} y = x + 2 \sqrt{3}$

$\tt \:  ⟼ \: x \: - \: \sqrt{3} y + 2 \sqrt{3} = 0$

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